Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on ${\mathbb{R}}^2$ with vorticity in the real Hardy space $H^p({\mathbb{R}}^2)$. In the present paper, we develop tools that significantly improve that result in two ways: Firstly, we achieve vorticities in $H^p({\mathbb{R}}^2)$ in the optimal range $p\in (0,1)$ compared to $(2\mathrm{/}3,1)$ in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.